Properties

Label 30T45
Order \(180\)
n \(30\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_3\times A_5$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $45$
Group :  $C_3\times A_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8)(2,9)(3,7)(10,29)(11,30)(12,28)(13,25)(14,26)(15,27)(16,24)(17,22)(18,23), (1,7,15,30,18,2,8,13,28,16,3,9,14,29,17)(4,25,22,20,10,5,26,23,21,11,6,27,24,19,12)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
60:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 5: None

Degree 6: None

Degree 10: $A_{5}$

Degree 15: None

Low degree siblings

15T15 x 2, 15T16, 18T90, 36T176, 45T16

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $20$ $3$ $( 4,11,14)( 5,12,15)( 6,10,13)( 7,18,25)( 8,16,26)( 9,17,27)(19,23,29) (20,24,30)(21,22,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 4,20)( 5,21)( 6,19)(10,29)(11,30)(12,28)(13,23)(14,24)(15,22)(16,26)(17,27) (18,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 2, 3)( 4,12,13)( 5,10,14)( 6,11,15)( 7,16,27)( 8,17,25)( 9,18,26) (19,24,28)(20,22,29)(21,23,30)$
$ 6, 6, 6, 6, 3, 3 $ $15$ $6$ $( 1, 2, 3)( 4,21, 6,20, 5,19)( 7, 8, 9)(10,30,12,29,11,28)(13,24,15,23,14,22) (16,27,18,26,17,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 3, 2)( 4,10,15)( 5,11,13)( 6,12,14)( 7,17,26)( 8,18,27)( 9,16,25) (19,22,30)(20,23,28)(21,24,29)$
$ 6, 6, 6, 6, 3, 3 $ $15$ $6$ $( 1, 3, 2)( 4,19, 5,20, 6,21)( 7, 9, 8)(10,28,11,29,12,30)(13,22,14,23,15,24) (16,25,17,26,18,27)$
$ 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 4, 8,16,24)( 2, 5, 9,17,22)( 3, 6, 7,18,23)(10,25,19,13,29) (11,26,20,14,30)(12,27,21,15,28)$
$ 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 4,26,16,20)( 2, 5,27,17,21)( 3, 6,25,18,19)( 7,23,10,29,13) ( 8,24,11,30,14)( 9,22,12,28,15)$
$ 15, 15 $ $12$ $15$ $( 1, 5, 7,16,22, 3, 4, 9,18,24, 2, 6, 8,17,23)(10,26,21,13,30,12,25,20,15,29, 11,27,19,14,28)$
$ 15, 15 $ $12$ $15$ $( 1, 5,25,16,21, 3, 4,27,18,20, 2, 6,26,17,19)( 7,24,12,29,14, 9,23,11,28,13, 8,22,10,30,15)$
$ 15, 15 $ $12$ $15$ $( 1, 6, 9,16,23, 2, 4, 7,17,24, 3, 5, 8,18,22)(10,27,20,13,28,11,25,21,14,29, 12,26,19,15,30)$
$ 15, 15 $ $12$ $15$ $( 1, 6,27,16,19, 2, 4,25,17,20, 3, 5,26,18,21)( 7,22,11,29,15, 8,23,12,30,13, 9,24,10,28,14)$

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [180, 19]
Character table:   
      2  2  .  2  2   .   2  2   .   2  .  .   .   .   .   .
      3  2  2  1  2   2   1  2   2   1  1  1   1   1   1   1
      5  1  .  .  1   .   .  1   .   .  1  1   1   1   1   1

        1a 3a 2a 3b  3c  6a 3d  3e  6b 5a 5b 15a 15b 15c 15d
     2P 1a 3a 1a 3d  3e  3d 3b  3c  3b 5b 5a 15d 15c 15b 15a
     3P 1a 1a 2a 1a  1a  2a 1a  1a  2a 5b 5a  5b  5a  5b  5a
     5P 1a 3a 2a 3d  3e  6b 3b  3c  6a 1a 1a  3d  3d  3b  3b
     7P 1a 3a 2a 3b  3c  6a 3d  3e  6b 5b 5a 15b 15a 15d 15c
    11P 1a 3a 2a 3d  3e  6b 3b  3c  6a 5a 5b 15c 15d 15a 15b
    13P 1a 3a 2a 3b  3c  6a 3d  3e  6b 5b 5a 15b 15a 15d 15c

X.1      1  1  1  1   1   1  1   1   1  1  1   1   1   1   1
X.2      1  1  1  A   A   A /A  /A  /A  1  1   A   A  /A  /A
X.3      1  1  1 /A  /A  /A  A   A   A  1  1  /A  /A   A   A
X.4      3  . -1  3   .  -1  3   .  -1  E *E   E  *E   E  *E
X.5      3  . -1  3   .  -1  3   .  -1 *E  E  *E   E  *E   E
X.6      3  . -1  B   .  -A /B   . -/A  E *E   F   G  /F  /G
X.7      3  . -1  B   .  -A /B   . -/A *E  E   G   F  /G  /F
X.8      3  . -1 /B   . -/A  B   .  -A  E *E  /F  /G   F   G
X.9      3  . -1 /B   . -/A  B   .  -A *E  E  /G  /F   G   F
X.10     4  1  .  4   1   .  4   1   . -1 -1  -1  -1  -1  -1
X.11     4  1  .  C   A   . /C  /A   . -1 -1  -A  -A -/A -/A
X.12     4  1  . /C  /A   .  C   A   . -1 -1 -/A -/A  -A  -A
X.13     5 -1  1  5  -1   1  5  -1   1  .  .   .   .   .   .
X.14     5 -1  1  D  -A   A /D -/A  /A  .  .   .   .   .   .
X.15     5 -1  1 /D -/A  /A  D  -A   A  .  .   .   .   .   .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
C = 4*E(3)^2
  = -2-2*Sqrt(-3) = -2-2i3
D = 5*E(3)^2
  = (-5-5*Sqrt(-3))/2 = -5-5b3
E = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
F = -E(15)^7-E(15)^13
G = -E(15)-E(15)^4